cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A139247 Triangle read by rows: row n lists the divisors of n-th perfect number A000396(n) that are multiples of n-th Mersenne prime A000668(n).

Original entry on oeis.org

3, 6, 7, 14, 28, 31, 62, 124, 248, 496, 127, 254, 508, 1016, 2032, 4064, 8128, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896, 4193792, 8387584, 16775168, 33550336, 131071, 262142, 524284, 1048568, 2097136, 4193792
Offset: 1

Views

Author

Omar E. Pol, Apr 22 2008

Keywords

Comments

Also, row n list the divisors of n-th perfect number that are not powers of 2.
First term of row n is the n-th Mersenne prime A000668(n). Last term of row n is the n-th perfect number A000396(n). Row n has A000043(n) terms. The sum of row n is equal to A133049(n), the square of n-th Mersenne prime A000668(n).

Examples

			Triangle begins:
  3, 6,
  7, 14, 28
  31, 62, 124, 248, 496
  127, 254, 508, 1016, 2032, 4064, 8128
  ...
==========================================================
Row .... First term ..... Last term ....... Row sum ......
n ..... (A000668(n)) ... (A000396(n)) ... (A000668(n)^2) .
==========================================================
1 ............ 3 .............. 6 ......... 3^2 = 9
2 ............ 7 ............. 28 ......... 7^2 = 49
3 ........... 31 ............ 496 ........ 31^2 = 961
4 .......... 127 ........... 8128 ....... 127^2 = 16129
5 ......... 8191 ....... 33550336 ...... 8191^2 = 67092481

Links

Crossrefs

Cf. A000043, A000396, A000668, A018254, A018487, A133024, A133025, A133031, A133049, A135652, A135653, A135654, A135655.

A138882 Triangle read by rows: row n lists divisors of n-th even superperfect number A061652(n).

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 32, 64, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384
Offset: 1

Views

Author

Omar E. Pol, Apr 11 2008

Keywords

Comments

The number of divisors of n-th even superperfect number is equal to A000043(n), then row n has A000043(n) terms.
The sum of divisors of n-th even superperfect number is equal to n-th Mersenne prime A000668(n), then n-th row sum is equal to A000668(n).

Examples

			Triangle begins:
  1, 2
  1, 2, 4
  1, 2, 4, 8, 16
  1, 2, 4, 8, 16, 32, 64
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096
  ...
==============================================================
..... Mersenne ..............................................
....... prime ...............................................
n ... A000668(n) = Sum of divisors of A061652(n) .............
==============================================================
1 ........ 3 ... = 1+2
2 ........ 7 ... = 1+2+4
3 ....... 31 ... = 1+2+4+8+16
4 ...... 127 ... = 1+2+4+8+16+32+64
5 ..... 8191 ... = 1+2+4+8+16+32+64+128+256+512+1024+2048+4096

Links

Crossrefs

Cf. A000005, A000043, A000203, A000668, A019279, A061652, A133031.

Programs

  • Mathematica
    Flatten[Divisors[2^(MersennePrimeExponent[Range[7]]-1)]] (* Harvey P. Dale, Apr 28 2022 *)

A138881 Array read by rows: row n lists divisors of n-th positive triangular number A000217(n).

Original entry on oeis.org

1, 1, 3, 1, 2, 3, 6, 1, 2, 5, 10, 1, 3, 5, 15, 1, 3, 7, 21, 1, 2, 4, 7, 14, 28, 1, 2, 3, 4, 6, 9, 12, 18, 36, 1, 3, 5, 9, 15, 45, 1, 5, 11, 55, 1, 2, 3, 6, 11, 22, 33, 66, 1, 2, 3, 6, 13, 26, 39, 78, 1, 7, 13, 91, 1, 3, 5, 7, 15, 21, 35, 105
Offset: 1

Views

Author

Omar E. Pol, Apr 11 2008

Keywords

Examples

			Array begins:
1
1, 3
1, 2, 3, 6
1, 2, 5, 10
1, 3, 5, 15
1, 3, 7, 21
1, 2, 4, 7, 14, 28

Crossrefs

Cf. A000217, A133021, A133031, A138882.

A139246 Triangle read by rows: row n lists the proper divisors of n-th perfect number A000396(n).

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 7, 14, 1, 2, 4, 8, 16, 31, 62, 124, 248, 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896, 4193792, 8387584, 16775168, 1
Offset: 1

Views

Author

Omar E. Pol, Apr 22 2008, corrected Apr 25 2008

Keywords

Comments

Rows n has A133033(n) terms.
The n-th row sum is the n-th perfect number A000396(n).

Examples

			Triangle begins:
  1, 2, 3
  1, 2, 4, 7, 14
  1, 2, 4, 8, 16, 31, 62, 124, 248
  1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064
  ...

Links

Crossrefs

Cf. A000396, A018254, A018487, A133024, A133025, A133031, A133033, A135652, A135653, A135654, A135655, A139247.

Programs

  • Mathematica
    Table[Most[Divisors[PerfectNumber[n]]],{n,6}]//Flatten (* Harvey P. Dale, Jul 08 2024 *)

A139248 Triangle read by rows: row n lists the proper divisors of n-th even superperfect number A061652(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 8, 1, 2, 4, 8, 16, 32, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 1, 2, 4, 8, 16
Offset: 1

Views

Author

Omar E. Pol, Apr 22 2008

Keywords

Comments

Also, row n list the proper divisors of n-th superperfect number A019279(n), if there are no odd superperfect numbers.
Row n has A000043(n) - 1 = A090748(n) terms.

Examples

			Triangle begins:
  1
  1, 2
  1, 2, 4, 8
  1, 2, 4, 8, 16, 32
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768
  ...

Links

Crossrefs

Cf. A000043, A000668, A018254, A018487, A019279, A061652, A090748, A133024, A133025, A133031, A133049, A135652, A135653, A135654, A135655, A139246.

A269065 Irregular triangle read by rows: row n lists divisors of n-th composite number.

Original entry on oeis.org

1, 2, 4, 1, 2, 3, 6, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 2, 3, 4, 6, 12, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 2, 3, 6, 9, 18, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 2, 3, 4, 6, 8, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 4, 7, 14, 28, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 4, 8, 16, 32, 1, 3, 11, 33, 1, 2, 17, 34
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 21 2016

Keywords

Comments

Subsequence of A027750.
Row sums give A073255.
Right border gives A002808.

Examples

			Triangle begins:
1,  2,  4;
1,  2,  3,  6;
1,  2,  4,  8;
1,  3,  9;
1,  2,  5,  10;
1,  2,  3,  4,  6,  12;
1,  2,  7,  14;
1,  3,  5,  15
1,  2,  4,  8,  16;
1,  2,  3,  6,  9,  18;
1,  2,  4,  5,  10, 20;
1,  3,  7,  21;
1,  2,  11, 22;
1,  2,  3,  4,  6,  8,  12, 24;
1,  5,  25;
1,  2,  13, 26;
1,  3,  9,  27;
1,  2,  4,  7,  14, 28;
1,  2,  3,  5,  6,  10, 15, 30;
1,  2,  4,  8,  16, 32;
1,  3,  11, 33;
1,  2,  17, 34;
...

Links

Crossrefs

Cf. A002808, A027750, A035004 (row length), A133021, A133031, A138881.

Programs

  • Mathematica
    Flatten[Table[Divisors[Composite[n]], {n, 22}]]
  • PARI
    tabf(nn) =  forcomposite(c=1, nn, print(divisors(c), ", ")); \\ Michel Marcus, Feb 21 2016
Showing 1-6 of 6 results.

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